Solve 1986/sqrt{1997-1}

$$\frac { 1986 } { \sqrt { 1997 } - 1 }$$

Evaluate

$$\frac{993\left(\sqrt{1997}+1\right)}{998}\approx 45.45890912$$

Solution Steps

Rationalize the denominator of \frac{1986}{\sqrt{1997}-1} by multiplying numerator and denominator by \sqrt{1997}+1.

$$\frac{1986\left(\sqrt{1997}+1\right)}{\left(\sqrt{1997}-1\right)\left(\sqrt{1997}+1\right)}$$

Consider \left(\sqrt{1997}-1\right)\left(\sqrt{1997}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{1986\left(\sqrt{1997}+1\right)}{\left(\sqrt{1997}\right)^{2}-1^{2}}$$

Square \sqrt{1997}. Square 1.

$$\frac{1986\left(\sqrt{1997}+1\right)}{1997-1}$$

Subtract 1 from 1997 to get 1996.

$$\frac{1986\left(\sqrt{1997}+1\right)}{1996}$$

Divide 1986\left(\sqrt{1997}+1\right) by 1996 to get \frac{993}{998}\left(\sqrt{1997}+1\right).

$$\frac{993}{998}\left(\sqrt{1997}+1\right)$$

Use the distributive property to multiply \frac{993}{998} by \sqrt{1997}+1.

$$\frac{993}{998}\sqrt{1997}+\frac{993}{998}$$